You are asking two questions.
Question #1: "Did Sidious train with Vader?"
Answer #1: Yes, he did. He trained not only with Vader, but dozens of dark side adepts.
Question #2: "Did Sidious do any training himself?"
Answer #2: Yes, he did. When the Emperor trained with others, not only is it logical that his own abilities would improve - but they would be active and in fighting condition. Likewise, Vader: the Complete Visual Guide confirms that Sidious allowed people such as Moff Tarkin and Sate Pestage to take care of the more tedious activities so that he could dedicate himself completely to the dark side. There have been accounts of him visiting Korriban, interrogating captured Jedi, raiding the Temple, visiting various Sith temples and points of interest, and utilizing his galaxy-wide resources to further his studies.
Really, Darthsith, it is a rather stupid assertion that he didn't improve.